Theorem prnz 4738

Description: A pair containing a set is not empty. (Contributed by NM, 9-Apr-1994.)

Hypothesis

Ref	Expression
prnz.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
prnz	⊢ {𝐴, 𝐵} ≠ ∅

Proof of Theorem prnz

Step	Hyp	Ref	Expression
1		prnz.1	. . 3 ⊢ 𝐴 ∈ V
2	1	prid1 4723	. 2 ⊢ 𝐴 ∈ {𝐴, 𝐵}
3	2	ne0ii 4293	1 ⊢ {𝐴, 𝐵} ≠ ∅

This proof depends on syntax axioms:   ∈ wcel 2147   ≠ wne 2959  Vcvv 3454  ∅c0 4281  {cpr 4590

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-v 3456  df-dif 4050  df-un 4154  df-nul 4282  df-sn 4589  df-pr 4591

This theorem is used by:  opnz  5446  propssopi  5484  fiint  9346  wilthlem2  27426  upgrbi  29701  wlkvtxiedg  30235  shincli  31994  chincli  32092  constrextdg2lem  34450  spr0nelg  48763  sprvalpwn0  48770